# Nanophotonics and Metrology laboratory NAM

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### Modelling

Our main interest is the interaction of electromagnetic fields with small dimensions systems (small compared to the wavelength, so in the optical regime, the system we are interested in range from a few nanometers to several micrometers). Over the years we have developped very sophisticated modelling techniques based on the integral formulation of Maxwell's equations and utilizing the formalism of the Green's tensor. These methods are particularily well suited for the modelling of geometries that incorporate scatterers embedded in stratified backgrounds, as illustrated in the following figure.

This figure shows the evolution of the modelling techniques developed at the NAM: (a) scatterer in infinite homogeneous space, (b) scatterer on a surface, (c) scatterer in a stratified medium. This last situation is illustrated in (d), where we show the computation of the interaction between a gold nanoparticles and a gold thin film, leading to a very strong field enhancement at their interface. For that 3D calculation, only the particle volume had to be discretized; the rest of the system - including the complex stratified background with the thin metal film - was handled semi-analytically using the corresponding Green's tensor.

More recently, we have moved one step further by developping a surface integral technique that requires only the surface of the scatterer to be discretized. This dramaticaly increase the performance of the technique since the numerical requirements scale with the surface of the scatterer, not its volume anymore. Furthermore, using a triangular mesh and finite elements basis functions that already partly fulfill Maxwell's equations, we were able to handle analytically the singularity of the Green's tensor, hence producing a very stable algorithme. Furthermore, this approach can be used to compute the field at very short distances from the scatterer, which is important to investigate effects such as plasmon-enhance Raman scattering of fluorescence. Finally, the triangular finite elements mesh makes possible the modelling of realistic scatterers, such as the plasmonic antenna shown in the following figure, which discretized geometry (middle) was derived from the electron microscopy image on the left. The right part of the figure shows the field intensity in and around the antenna when it is excited at its resonance.

The most recent development is a technique based on the surface integral representation that include a periodic Green's tensor. This way it is possible to compute light scattering in complex geometries in 1, 2 and 3 dimensions, by merely discretizing the surface of a single scatterer, as illustrated in the following figure which shows a typical example where light is scattered scattered by a photonic crystal made with an infinite square array of pillars with a refractive index of 3.36 in air. The real part of the total (incident+scattered) instantaneous electric field is calculated in planes at 500 nm above, 500 nm below, and in the array for a 45° p-polarized plane wave incident from above. The scale is normalized in each frame. The arrow length is proportional to the electric field. (a),(b) No substrate; (c),(d) with substrate. Different illumination wavelengths λ are considered: (a) λ=350 nm, (b) λ=700 nm, (c) λ=340 nm, (d) λ=700 nm.

All these techniques are described in our publications. Some programs are available in the download section of this website and we are currently working on making the most recent programs freely available to the community.